Lecture 16

Recall from last lecture:

The Yukawa Theory of Quantum Exchange

This section and the next present a "hand waving" argument for the form of the propagator expression. The following is my summary of the argument.

The binding of protons and neutrons in a nucleus was postulated to be due to a short range strong force with a potential of the form:

U(r) = (g₀/4pr)e^-r/R

where R can be related to a mass by

R = hbar/mc.

Without the exponential term, the form of the potential is like the electrostatic potential U = Q/4pr, leading us to interpret g₀ as a sort of "strong force charge". Now, we simply call g₀ the strong coupling constant. The mass in R is interpreted as the mass of the boson exchanged between two particles, and, for m>0, it makes the force short range. Yukawa's theory now is seen as an approximation to the true effects of the strong force with massless gluons.

The Boson Propagator

This idea resurfaces in modern quantum field theory. Consider the interaction of a particle with a heavy source of force. The electron and nucleus exchange a boson, altering the electron's momentum by an amount q. The probability for momentum transfer q is just the Fourier transform, f(q), of the potential U(r). (Demonstration of this is covered in the field theory course.)

f(q) = g Re{Int[U(r) e^iq·r dV]}

where the electron couples to the boson with strength g. For a central potential, U(r) = U(r), and (after a few steps) the integration yields

f(q) = gg₀ /(|q|² + m²)

This is the momentum space description of the interaction with a heavy source that couples to the boson with strength g₀.

We used a heavy source so that we didn't have to worry about the recoil of the source, and it made the integral non-relativistic. In the general case, we need an expression with a covariant form, and we get that by changing |q|² to the four vector squared:

f(q²) = g g₀ / (m² - q²)

(Note that the difference in sign with Eq. 2.6 in Perkins is due to the different relativistic metric.) This expression says that the probability for two particles to exchange a boson with 4-momentum transfer q² is given by the product of the boson couplings to the particles divided by the difference between the mass squared of the (real) boson and the q². When the q² equals m², the probability becomes large. For photons and gluons, this occurs for q²=0. For W^± and Z°, this occurs around q²=90GeV!